246 IEEE GEOSCIENCE AND REMOTE SENSING LETTERS, VOL. 4, NO. 2, APRIL 2007

Quartic-Phase Algorithm for Highly SquintedSAR Data ProcessingKaizhi Wang and Xingzhao Liu, Member, IEEE

Abstract—In this letter, an algorithm based on a quartic-phasemodel is discussed for processing highly squinted synthetic aper-ture radar (SAR) data from a large range swath. In the algo-rithm, a precise quartic-phase model is adopted to describe arange-dependent property of the SAR signal; a constant factor anda secondary scaling process are introduced to make the algorithmeasy to be utilized compared with traditional nonlinear chirpscaling algorithms. The novel algorithm can process SAR dataunder a squint angle above 50◦ and achieve a focus depth over60 km.

Index Terms—Quartic-phase algorithm, radar imaging,squinted synthetic aperture radar.

I. INTRODUCTION

IN CONVENTIONAL synthetic aperture radar (SAR) sys-tems, the antenna is pointed at broadside, i.e., the pointing

direction of the antenna is nearly perpendicular to the flightpath [1]. When there is an offset angle between the pointingdirection of the antenna and broadside, the SAR system isworking under squint mode and the offset angle is the so-calledsquint angle. Squint mode working can increase the flexibilityof a SAR system and gather more information about a scene.The squint working mode can also be applied in a small satelliteSAR system and constellation system, which is composed ofseveral independent SAR sensors distributed in the space, andeach of them works under a specific squint angle.

There are a number of algorithms developed for SAR imag-ing, such as range Doppler (R-D) algorithms [2], [3], ω−kalgorithms [4], [5] (ω−k), chirp-scaling (CS) algorithm [6],extended-CS (ECS) algorithms [7], [8], nonlinear-CS (NCS)[9]–[12] algorithms, extended-exact-transfer-function (EETF)algorithms [13]–[16] and the algorithms of subaperture type[17], [18], etc. In those algorithms, a focusing filter is con-structed at a reference range, and the data at this rangecan be focused perfectly. To yield a large depth of focus, arange-variant processing, which is designed according to someapproximation of range-azimuth coupling term, is used to com-pensate the difference between signals at the reference rangeand the other range cells.

In the algorithms of R-D, ω−k, and CS, the coupling termis approximated by a quadratic function of fr, the range fre-quency, in 2-D frequency domain. The range-variant chirp rate,

Manuscript received September 21, 2006; revised October 27, 2006.The authors are with the Department of Electronic Engineering, Shanghai

Jiaotong University, Shanghai 200240, China.Digital Object Identifier 10.1109/LGRS.2006.890552

i.e., the coefficient of the quadratic term, is simply assumedto be a constant in the whole scene. The shape of the range-migration curve is range dependent, and a linear function ofrange is adopted to describe the variation in range-Dopplerdomain. With a process of interpolation or CS in range-Dopplerdomain, the difference of migration curves at various rangecells can be eliminated. Since assuming the range-variant chirprate as a constant, the algorithms of R-D, ω−k, and CS typehave a small depth of focus and those algorithms may notbe applicable to process highly squinted SAR data for theirsimple approximation to a SAR system.

In the ECS algorithm, a higher order model of the couplingterm between range and azimuth is adopted. The coupling termis expanded into a cubic or a higher order polynomial of fr. Thehigh-order terms (above second) of the polynomial are removedat reference range in the 2-D frequency domain. However, suchan operation can only reduce the effect of those terms for theirrange-dependent property. Although a more precise model isadopted in ECS, the improvement of performance is limitedsince the approximation ignores the range-dependent propertyof high-order terms in the expansion.

In the EETF algorithms, the transfer function of a SARsystem is deduced at a reference range under some assumptions.The SAR data are focused by a process of antifiltering. EETFalso considers the difference of azimuth signal at various rangecells, and the difference is compensated by a range-dependentfilter in the range-Doppler domain. EETF does not take therange-variant property of a SAR system into account whenconstructing the transfer function; therefore the performance ofEETF is the best at the reference range and degrades quicklyat the range cells apart from the reference range. A fourth-order EETF algorithm (EETF4) is proposed and discussed in[13]–[16]. It is demonstrated that a fourth-order model of phaseis necessary for ultrahigh-resolution SAR processing.

There are also algorithms of subaperture type for squintedSAR data processing. In those algorithms, a full-aperture timeis divided into several subapertures so as to avoid large rangemigration and make the focusing algorithm easy to be designed.But the algorithm needs an extra step to combine the resultsfrom each subaperture into a full-aperture image.

NCS is an effective algorithm to focus highly squinted SARdata. In the algorithm of [9], a cubic polynomial of fr is adoptedto model the coupling term in the 2-D frequency domain.The cubic term of fr is removed at the reference range. Therange-dependent chirp rate is represented by a linear functionof range. The range-variant property of the migration curveand the chirp rate are eliminated by an NCS operation in therange-Doppler domain. Due to the elaborate structure, NCS has

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WANG AND LIU: QUARTIC-PHASE ALGORITHM FOR HIGHLY SQUINTED SAR DATA PROCESSING 247

Fig. 1. Geometric relationship of the squint mode SAR.

the ability to process highly squinted SAR data [9], [11]. Butthere are still some shortcomings. First, an azimuth referencefrequency should be chosen as a parameter of the algorithm.The frequency not only depends on the system parameters butalso varies with the squint angle. Besides, the output image isscaled in range direction. The scaling quantity relates with thechoice of the reference frequency.

Inspired by NCS and EETF4, a quartic-phase algorithm(QPA) is proposed in this letter so as to combine the advantagesand overcome shortcomings of the two algorithms. In the QPA,the phase term of SAR data from a point target is described bya cubic polynomial of range frequency or fast time (i.e., rangetime) in 2-D frequency domain or range-Doppler domain. Thecoefficients of the polynomial are represented by functions ofrange. A constant factor β is introduced to replace the referencefrequency defined in NCS. β is invariant with the squint angle.A secondary CS process is employed to remove the scalingeffect in range completely [7]. The new algorithm has theability to process the SAR data with large squint angle froma wide range swath.

This letter is organized as following: the geometric rela-tionship and signal model of squinted SAR are described inSection II; the algorithm is discussed in Section III; choiceof β, the constant scaling factor, is analyzed in Section IV; somesimulation results are presented in Section V and conclusion isin Section VI.

II. GEOMETRY AND SIGNAL MODEL

A simple geometric illustration of a squinted SAR system isshown in Fig. 1. The spacecraft moves along the x-direction atan altitude of H with a constant velocity ν, which is a relativevelocity between the earth and the spacecraft. The radar antennatransmits and receives pulses in a squinted direction defined bythe squint angle θ and the look angle Φ.

Assuming that the closest slant distance from a point targetto the SAR platform at zero squint angle is R = H/cosΦ, theinstantaneous distance at slow time (azimuth time) t is

r(t) =√R2 + (t+ ts)2ν2 ts = R tan θ/ν. (1)

The range of t is from −T/2 to T/2, where T is aperture time.In Fig. 1, r(0) is the instantaneous slant range at t = 0, i.e., the

center of aperture. The SAR signal from a point target can beexpressed as

s(t, τ, R) = a(t, τ, R) exp

[−jπk

(τ − 2r(t)

c

)2

− j 4πr(t)λ

](2)

where a(t, τ, R) is the function responsible for all variation inamplitude, k is chirp rate of transmitted signal, τ is the fast time(range time), λ is wave length, and c is the velocity of light.With reference of [19] and [20], and omitting the amplitudefunction in (2), the SAR signal in the 2-D frequency domaincan be expressed as

S(fa, fr, R) = exp

−j 4πR

λ

√(1 +

frλ

c

)2

−(faλ

2ν

)2

× exp(jπf2

r

k+ j2πfats

)(3)

where fr and fa are the frequency of range and azimuth,respectively. The phase term of (3) can be expended into a cubicpolynomial of fr as

(2πfats −

4πR√A

λ

)− 2πfrτR +

πf2r

KmR+ πKcRf

3r (4)

where

A =1 −(faλ

2ν

)2

τR =2Rc√A

KmR =k

1 +Rkγ1γ1 =

2λ(1 −A)c2(

√A)3

KcR =Rγ2 γ2 =2λ2(A− 1)c3(

√A)5

. (5)

The subscript “R” of KmR, KcR, and τR denotes their valuesat range R. The phase term in (4) can be divided into two parts:the coupling of azimuth and range

exp[j

(−2πfrτR +

πf2r

KmR+ πKcRf

3r

)](6)

and the residual part

exp

(j2πfats − j

4πR√A

λ

). (7)

With reference to the methods described in [10], [19], and [20],the coupling term in (6) can be transformed into the range-Doppler domain as

exp{−jπ

[KmR(τ − τR)2+KcR(KmR)3(τ − τR)3

]}. (8)

248 IEEE GEOSCIENCE AND REMOTE SENSING LETTERS, VOL. 4, NO. 2, APRIL 2007

The variation of Km and Kc with range R can be described byfollowing functions:

KmR = Kmref + ks1∆τ + ks2∆τ2

KcR = Kcref + ks3∆τ (9)

where ∆τ = (2(R−Rref))/(c√A), Rref is the reference

range, Kmref and Kcref are the values of Km and Kc at Rref ,and ks1, ks2, and ks3 are expressed as follows:

ks1 = − γ1Km2refc

√A/2

ks2 =(γ1c

√A)2

Km3ref/4

ks3 =12c√Aγ2. (10)

III. ALGORITHM DESCRIPTION

To focus the SAR data perfectly in the whole scene, the rangedependence of data should be removed. In the algorithms ofR-D, ω−k, and CS, the difference of range migration curves atvarious range cells is considered and eliminated by scaling thecurves in range-Doppler domain to a same shape. In NCS, notonly the migration curve but also the variation of Km with rangeis taken into account by a linear function of range. Comparedwith NCS, a more elaborated model is utilized in the novelalgorithm to describe range dependent property of SAR dataand a better performance is expected.

A similar method as stated in [6] and [9] is taken to reshapemigration curves at various range cells to a same shape. Theprocedure can be concluded into two steps.

1) A reference range should be selected and the correspond-ing migration curve (i.e., τref ) is regarded as a standard,that is, all the other curves (such as τR) will be scaled tothe shape of the standard one.

2) The migration curve τR is a function of Doppler fre-quency. If the distance of two curves is equal at anyDoppler frequency, the two curves will share a sameshape. The distance at Doppler centroid can be selectedas a reference and the distance at other Doppler frequencywill be scaled to the reference by multiplying with afactor α, which is a function of Doppler frequency.

The scaling factor can be expressed as

α(fa) =

√1 −

(λfa2ν

)2

√1 −

(λfdc2ν

)2. (11)

Let τs be the reshaped version of τR. The relationship betweenτref , τR, τs, and ∆τ can be described by

τref = τs − α∆τ

τR = τs − (α− 1)∆τ. (12)

The scaling process can be carried out by a multiplicationwith a chirp signal in the range-Doppler domain [6]. Besides the

scaling of migration curves, another objective is to eliminate therange dependent property of SAR data. The chirp function willbe an oscillation signal of quartic phase and range independent;therefore the model takes a form of

CS1 = exp[−jπp1(τ − τref)2 − jπp2(τ − τref)3

]× exp

[−jπp3(τ − τref)4

]. (13)

To yield the expressions of p1, p2, and p3, two parameters,referred as Y1 and Y2, will be introduced by filtering the signalwith respect to fr in the 2-D frequency domain with

Mpre = exp[jπ(Y1f

3r + Y2f

4r

)]. (14)

The CS result can be yielded by multiplying (8) with (13).To eliminate ∆τ , the coefficients of the terms, which contain∆τn(τ − τs)m, m �= 0, n �= 0 should be set to zero. A factshould be noticed that Y1 and Y2 are always multiplied by(α− 1). To avoid the solved Y1 and Y2 being infinitive whenα = 1, the scaling factor α(fa) is multiplied with a constantfactor β to make it far away from unit in the azimuth bandwidth.The solved parameters are

p1 = Kmref(1−βα)

βα

p2 = 1−βα3βα ks1

p3 = − 2ks2+9Y1Km2refks1(βα−1)+3ks3Km3

ref(βα−1)

12βα

Y1 = βα−23(βα−1)Km3

refks1

Y2 = 4p3βα+3Y1Km2refks1+ks3Km3

ref4Km4

ref(βα−1)

. (15)

The result of CS can be divided into two parts: the terms about(τ − τs) for imaging

exp{−jπ

[(Km + p1)(τ − τs)2 +

(Y1Km3 + p2

)(τ − τs)3

+(p3 − Y2Km4

)(τ − τs)4

]}(16)

and the remainder phase of CS. By removing the high-order(above 2) phase term of fr in the 2-D frequency domain, theSAR data are ready to be focused in range, shown as

exp{−jπ(Km + p1)(B − βα∆τ)2

}B=τ − τref . (17)

To remove the constant factor β, a second CS procedure isemployed. The CS function is

CS2 = exp[−jπ(Kmref + p1)(β − 1)B2

]. (18)

Result of the second CS algorithm is

exp[jπ

αf2r

Kmref− j2π(τref + α∆τ)fr

]. (19)

Removal of range curve and focusing in range direction are car-ried out by filtering chirp-scaled data (19) in the 2-D frequencydomain with function of

MR = exp(−jπ αf2

r

Kmref

)+ j2π

(τref −

2Rref

c cos θfr

)(20)

WANG AND LIU: QUARTIC-PHASE ALGORITHM FOR HIGHLY SQUINTED SAR DATA PROCESSING 249

Fig. 2. Flow chart of QPA. conj means a conjugation operation on phr.

and the range-focused SAR data

exp

[−j2π

(2R

√A

λ− fats +

2Rc cos θ

fr

)](21)

is yielded. An inverse Fourier transform takes (21) into range-Doppler domain and ready to be focused in azimuth direction.A range-variant filter

MA = exp

[j4πR

√A

λ

](22)

will be applied to focus SAR data in azimuth direction. Thefilter can be updated for each or several range cells to generatean acceptable SAR image. In time domain, the focused pointtarget is located at 2R/(c cos θ) in range and R tan θ/ν inazimuth. Fig. 2 shows the flow chart of QPA.

There is also a residual phase term from the two scalingprocedures, and it is expressed as

phr = exp[jπ(a2∆τ2 + a3∆τ3 + a4∆τ4

)]

× exp

{−jπ(Km + p1)β(β − 1)

[2(R−Rref)c cos θ

]2}(23)

where

a2 = Km(βα− 1)2 + (βα)2p1

a3 = ks1(βα− 1)2 + Y1Km3(βα− 1)3 + p2(βα)3

a4 = ks2(βα−1)2+3Y1Km2ks1(βα−1)3−Y2Km4(βα−1)4

+ Km3ks3(βα− 1)3 + p3(βα)4. (24)

TABLE ISIMULATION PARAMETERS

Fig. 3. Simulation results under squint angle 50◦.

IV. CHOICE OF β

Being a ratio, the scaling factor α in (11) is nearly stableunder various squint angles. The choice of β will affect thecorrection of the Fourier transform on the SAR signal. As forthe Fourier transform of polynomial phase signal, the second-order term is required to be the dominant part of the phase [19],[20]. Thus, the choice of |β| is “the greater, the better” becausethe denominator of the second-order phase term is linear with βwhile the denominators of the third order and the fourth orderare quadratic with β. It should be noticed that too large a |β|may make the bandwidth of (18) out of the sampling frequency.Thus, the principle of the choice of β is to make its absolutevalue as large as possible while keeping the bandwidth underthe sampling frequency. In fact, when β ≈ 1, the secondary CScan be omitted. Specially, when β = −1, the image will not bescaled other than an inverse in the range direction.

V. SIMULATIONS

In this section, some simulation results and comparisons withNCS [10] are presented to demonstrate performance of QPA.Some basic parameters for the simulations are listed in Table I.

The simulations are carried out under squint angles of 50◦,60◦, and 70◦. Four point targets are placed in the scene alongrange direction, and their slant ranges are 700 (i.e., center ofscene), 690, 670, and 650 km. The simulation results are shownin Figs. 3–5. Quality parameters of QPA simulation results areshown in Table II. Resolutions of range and azimuth of eachtarget are listed in the columns of RR and RA.

Theoretic resolution of range is 5.56 m for the simulations,while azimuth resolution is slant distance and squint angledependent and the theoretic values are 3.19, 6.78, and 21.20 mfor squint angles of 50◦, 60◦, and 70◦ at 700 km, respectively.There is no extra window functions added to SAR signal forside-lobe depression; therefore the theoretic value of peak side-lobe ratio (PSLR) will be about 13 dB and integrated side-loberatio (ISLR) will be about 10 dB.

250 IEEE GEOSCIENCE AND REMOTE SENSING LETTERS, VOL. 4, NO. 2, APRIL 2007

Fig. 4. Simulation results under squint angle 60◦.

Fig. 5. Simulation results under squint angle 70◦.

TABLE IISIMULATION RESULT OF TARGETS WITH QPA

In the simulations above, the reference range is set at700 km. It can be seen that the image quality of NCS degradeswhen target departs from reference range. The degradation isaccelerated when squint angle becomes large. QPA has a betterperformance than NCS and is able to process SAR data with asquint angle as large as 70◦. The image quality of QPA almoststays unchanged across range. The better performance is due toa range-variant model of the second and third phase term.

VI. CONCLUSION

The QPA is developed with reference to NCS and EETF4.A quartic-phase model and NCS operation are employed in thenovel algorithm and make it more suitable to process ultrahighresolution data of highly squinted SAR from a large rangeswath. A constant scaling factor is introduced to replace the

reference frequency in NCS, and this makes the algorithm easyto be used. An optional scaling process is also discussed toremove the effect from β, the constant scaling factor. The QPAcan achieve a satisfactory performance in processing of highlysquinted SAR data from a large range swath. If the optionalscaling process is not taken, the computation complexity ofthe new algorithm is slightly higher than that of NCS for theirsimilar steps in processing.

Of all the parameters for QPA, squint angle is the most sensi-tive parameter. Even a little error may cause image defocusingwhen the squint angle is large (above 45◦). The geometricmodel of SAR system is also very important for SAR data imag-ing. If the flight trajectory of SAR platform is unmatched withthe desired model, the output image will be degraded and thedegradation becomes worse when there is a large squint angle.

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